The SRB measures for infinite dimensional dynamical systems to be studied.
Li and Shu [1]} proved the Ruelle's inequality in Banach space and [2]} proved the equivalence of SRB measures and invariant measures satisfying the Pesin's entropy formula in Hilbert space for random dynamical systems.
The main difficulty in the extent of this result to Banach space is that there is no significant definition of volume and determination on Banach setting.
But recently, Blumenthal and Young [3]} overcome this difficulty, and proved that the equivalence of SRB measure and invariant measures satisfy the Pesin's entropy formula for a certain mapping on Banach spaces.
The more results about SRB measures for infinite dimensional dynamical systems can refer to [4]}, [5]}.
